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Khovanov-Rozansky Homology and Topological Strings

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Abstract

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks

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References

  1. Schwarz A. New topological invariants arising in the theory of quantized fields. Baku International Topological Conference Abstracts (part II) (1987)

  2. Witten E. (1989). Quantum field theory and the jones polynomial. Commun. Math. Phys. 121:351

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Freyd P., Yetter D., Hoste J., Lickorish W., Millett K., Oceanu A. (1985). A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12:239

    Article  MATH  MathSciNet  Google Scholar 

  4. Witten E. (1995). Chern-Simons gauge theory as a string theory. Prog. Math. 133:637 hep-th/9207094

    MathSciNet  Google Scholar 

  5. Gopakumar R., Vafa C. (1999). On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3:1415 hep-th/9811131

    MATH  MathSciNet  Google Scholar 

  6. Gopakumar, R., Vafa, C.: M-theory and topological strings. I,II. hep-th/9809187; hep-th/9812127

  7. Ooguri H., Vafa C. (2000). Knot invariants and topological strings. Nucl. Phys. B577:419

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Labastida J.M.F., Marino M., Vafa C. (2000). Knots, links and branes at large N. JHEP 0011:007 hep-th/0010102

    Article  ADS  MathSciNet  Google Scholar 

  9. Hosono S., Saito M.-H., Takahashi A. (1999). Holomorphic anomaly equation and BPS state counting of rational elliptic surface. Adv. Theor. Math. Phys. 3:177

    MATH  MathSciNet  Google Scholar 

  10. Hosono S., Saito M.-H., Takahashi A. (2001). Relative Lefschetz action and BPS state counting. Int. Math. Res. Notices 15:783

    Article  MathSciNet  Google Scholar 

  11. Katz S., Klemm A., Vafa C. (1999). M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3:1445 hep-th/9910181

    MATH  MathSciNet  Google Scholar 

  12. Maulik D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory, I. math.AG/0312059

  13. Katz, S.: Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds. math.ag/0408266

  14. Khovanov M. (2000). A categorification of the Jones polynomial. Duke Math. J. 101:359 math.QA/9908171

    Article  MATH  MathSciNet  Google Scholar 

  15. Khovanov M. (2005). Categorifications of the colored Jones polynomial. J. Knot Theory Ramifications 14:111 math.QA/0302060

    Article  MATH  MathSciNet  Google Scholar 

  16. Khovanov M. (2004). sl(3) link homology. Algebr. Geom. Topol. 4:1045 math.QA/0304375

    Article  MATH  MathSciNet  Google Scholar 

  17. Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. math.QA/ 0401268

  18. Bar-Natan D. (2002). On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2:337 math.QA/0201043

    Article  MATH  MathSciNet  Google Scholar 

  19. Khovanov, M.: An invariant of tangle cobordisms. math.QA/0207264

  20. Jacobsson M. (2004). An invariant of link cobordisms from Khovanov’s homology. Algebr. Geom. Topol. 4:1211 math.GT/0206303

    Article  MATH  MathSciNet  Google Scholar 

  21. Shumakovitch, A.: KhoHo – a program for computing and studying Khovanov homology. http://www.geometrie.ch/KhoHo

  22. Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. hep-th/0310272

  23. Labastida J.M.F., Marino M. (2001). Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217:423 hep-th/0004196

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Labastida, J.M.F., Marino, M.: A new point of view in the theory of knot and link invariants. math.qa/0104180

  25. Murakami H., Ohtsuki T., Yamada S. (1998). HOMFLY polynomial via an invariant of colored plane graphs. Enseign. Math. 44:325

    MATH  MathSciNet  Google Scholar 

  26. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. hep-th/0403166

  27. Kontsevich, M.: unpublished

  28. Kapustin A., Li Y. (2003). D-branes in Landau-Ginzburg models and algebraic geometry. JHEP 0312:005 hep-th/0210296

    Article  ADS  MathSciNet  Google Scholar 

  29. Orlov D. (2004). Triangulated categories of singularities and D-branes in Landau-Ginzburg orbifold. Proc. Steklov Inst. Math. 246:227 math.AG/0302304

    MathSciNet  Google Scholar 

  30. Brunner, I., Herbst, M., Lerche, W., Scheuner, B.: Landau-Ginzburg realization of open string TFT. hep-th/0305133

  31. Ashok, S.K., Dell’Aquila, E., Diaconescu, D.E.: Fractional branes in Landau-Ginzburg orbifolds. hep-th/0401135

  32. Ashok, S.K., Dell’Aquila, E., Diaconescu, D.E., Florea, B.: Obstructed D-branes in Landau-Ginzburg orbifolds. hep-th/0404167

  33. Hori K., Walcher J. (2005). F-term equations near Gepner points. JHEP 0501:008 hep-th/0404196

    Article  ADS  MathSciNet  Google Scholar 

  34. Brunner, I., Herbst, M., Lerche, W., Walcher, J.: Matrix factorizations and mirror symmetry: the cubic curve. hep-th/0408243

  35. Mirror symmetry (Clay mathematics monographs, V. 1). In: Hori, K. et al. (ed.) American Mathematical Society (2003)

  36. Schwarz, A., Shapiro, I.: Some remarks on Gopakumar–Vafa invariants. hep-th/0412119

  37. Taubes C. (2001). Lagrangians for the Gopakumar–Vafa conjecture. Adv. Theor. Math. Phys. 5:139 math.DG/0201219

    MATH  MathSciNet  Google Scholar 

  38. Dunfield, N., Gukov, S., Rasmussen, J.: The superpolynomial for knot homologies. math.GT/0505662

  39. Bar-Natan, D.: Some Khovanov-Rozansky Computations. http://www.math.toronto.edu/ drorbn/Misc/KhovanovRozansky/index.html

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Authors and Affiliations

  1. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Sergei Gukov & Cumrun Vafa

  2. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Albert Schwarz

Authors
  1. Sergei Gukov
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  2. Albert Schwarz
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  3. Cumrun Vafa
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Correspondence to Sergei Gukov.

Additional information

Mathematics Subject Classifications (2000): 57M25, 57M27, 18G60, 18E30, 14J32, 14N35, 81T30, 81T45

Dedicated to the memory of F.A. Berezin

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Gukov, S., Schwarz, A. & Vafa, C. Khovanov-Rozansky Homology and Topological Strings. Lett Math Phys 74, 53–74 (2005). https://doi.org/10.1007/s11005-005-0008-8

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  • DOI: https://doi.org/10.1007/s11005-005-0008-8

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